On the geometry of tangent hyperquadric bundles: CR and pseudoharmonic vector fields

We study the natural almost CR structure on the total space of a subbundle of hyperquadrics of the tangent bundle $T(M)$ over a semi-Riemannian manifold $(M,g)$ and show that if the Reeb vector $\xi$ of an almost contact Riemannian manifold is a CR map then the natural almost CR structure on $M$ is strictly pseudoconvex and a posteriori $\xi$ is pseudohermitian. If in addition $\xi$ is geodesic then it is a harmonic vector field. As an other application, we study pseudoharmonic vector fields on a compact strictly pseudoconvex CR manifold $M$, i.e. unit (with respect to the Webster metric associated with a fixed contact form on $M$) vector fields $X \in H(M)$ whose horizontal lift $X^\uparrow$ to the canonical circle bundle $S^1 \to C(M) \to M$ is a critical point of the Dirichlet energy functional associated to the Fefferman metric (a Lorentz metric on $C(M)$). We show that the Euler-Lagrange equations satisfied by $X^\uparrow$ project on a nonlinear system of subelliptic PDEs on $M$.